I'm a bit confused, I thought conformal inclusions apply to groups (like $SU(2)\subset SO(5)$) and not entire CFT's? Or is this a seperate definition?
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MichaelOct 14 '11 at 8:42

In VOA language, I would call a conformal inclusion a map $V\to W$ of VOAs that sends the Virasoro element of $V$ to the Virasoro element of $W$. But you're right, I've only seen the terminology used for the VOAs that correspond to loop groups.
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AndréOct 14 '11 at 12:54

For conformal nets $\mathcal A,\mathcal B,\ldots$ or $A,B,\ldots$ is typical. For Virasoro nets $\mathrm{Vir}_{c=\frac 12}$ is normally used and for loop group nets $\mathcal A_{G_k}$. In VOA it seems to be common to use
$V$
for a generic VOA. Kac uses in "VOA for Beginners" $V_Q$ for the lattice VOA associated with a lattice $Q$ and $V^k(\mathfrak g)$ for the affice VOA of $\mathfrak g$ at level $k$.